The first Grushin eigenvalue on cartesian product domains

Abstract

In this paper we consider the first eigenvalue ${\lambda }_1(\Omega )$ of the Grushin operator ${\Delta }_G:={\Delta }_{x_1}+|x_1{|}^{2s}{\Delta }_{x_2}$ with Dirichlet boundary conditions on a bounded domain $\Omega$ of ${\mathbb{R}}^d={\mathbb{R}}^{d_1+d_2}$. We prove that ${\lambda }_1(\Omega )$ admits a unique minimizer in the class of domains with prescribed finite volume which are the cartesian product of a set in ${\mathbb{R}}^{d_1}$ and a set in ${\mathbb{R}}^{d_2}$, and that the minimizer is the product of two balls ${\Omega }_1^{\ast }\subseteq {\mathbb{R}}^{d_1}$ and ${\Omega }_2^{\ast }\subseteq {\mathbb{R}}^{d_2}$. Moreover, we provide a lower bound for $|{\Omega }_1^{\ast }|$ and for ${\lambda }_1({\Omega }_1^{\ast }\times {\Omega }_2^{\ast })$. Finally, we consider the limiting problem as $s$ tends to $0$ and to $+\infty$.